In 43, 44, 45, 46, 47, 48, 49, and 50, the following definitions are used: A relation on a set A is defined to be irreflexive if, and only if, for every x € A‚ x K x; asymmetric if, and only if, for every x, y ≤ A if x Ry then y Kx; intransitive if, and only if, for every x, y, z € A, if x R y and y R z then x K z. For each of the relations in the referenced exercise, determine whether the relation is irreflexive, asymmetric, intransitive, or none of these. 43. R₁ = {(0,0), (0, 1), (0, 3), (1, 1), (1, 0), (2, 3), (3, 3)} Answer 44. R₂ = {(0, 0), (0, 1), (1, 1), (1, 2), (2, 2), (2, 3)} 45. R3 = {(2, 3), (3, 2)} Answer 46. R4 = {(1, 2), (2, 1), (1, 3), (3, 1)} 47. R5 = {(0,0), (0, 1), (0, 2), (1, 2)} 48. R6 = {(0, 1), (0, 2)}

14 Part 1 of 4 Required information NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the following relations: R₁ = {(a, b) = R² | a> b), the greater than relation R₂ = {(a, b) € R2 R3 = {(a, b) € R² | a < b), the less than relation R₁ = [(a, b) ER² | a ≤ b), the less than or equal to relation R5 = {(a, b) € R² | a= b), the equal to relation R₁ = {(a, b) = R² | a b), the unequal to relation For these relations on the set of real numbers, find R₂0 R2. Multiple Choice R4 R3 I az b), the greater than or equal to relation R2 R₁

5. Let X be the set of all subsets of Z. Define a relation R on X as follows: Given two subsets of integers A and B, ARB if and only if An BC = Ø. Is R an equivalence relation? Prove or disprove.

11. Suppose A = {a,b,c,d} and R = {(a, a),(b,b), (c,c), (d,d)}. Is R reflexive? Symmet- ric? Transitive? If a property does not hold, say why. 12. Prove that the relation | (divides) on the set Z is reflexive and transitive. (Use Example 16.8 as a guide if you are unsure of how to proceed.) 13. Consider the relation R = {(x, y) eRxR:x-ye Z} on R. Prove that this relation is reflexive, symmetric and transitive.

Consider the following two statements: (1) If a relation on a nonempty set is reflexive, then it is not irreflexive. (2) If a relation on a nonempty set is not reflexive, then it is irreflexive O Neither (1) nor (2) is always true. O Only (1) is always true. O Only (2) is always true. Both (1) and (2) are always true.